Dissertation Proposal
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Presentation for dissertation proposal: Effect of visual modeling on integer mapping
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Dissertation Proposal
- 1. Effect of Visual Modeling on Number Mapping Sara Stetson Rivier University Dissertation Proposal
- 2. Number Sense “It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two: the degree of abstraction required is far from easy.” ~ Bertrand Russel Introduction to Mathematical Philosophy (1919, p. 3)
- 3. What is Number Sense? • Infants • Animals • Indigenous People • Brain studies
- 4. Approximate Number System • Fight or Flight • Foraging vs.
- 5. Number Sense Number sense and the Approximate Number System are the same thing =
- 6. Approximate Number System • Modality invariance • Objects, sounds, actions (e.g., Brannon, 2001; Dehaene, et al., 2004; Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005; Xu, 2002)
- 7. Approximate Number System No tracking of individuals Approximate comparison, addition, and subtraction (e.g., Brannon, 2001; Xu, 2002; Lipton & Spelke, 2003; Wood & Spelke, 2005; McCrink & Wynn, 2005; deHevia& Spelke, 2008)
- 8. Approximate Number System Natural association with space Natural propensity to represent quantity on mental number line (deHevia& Spelke, 2008)
- 9. Approximate Number System (e.g., Brannon, 2001; Berger, et al., 2006; Libertus & Brannon, 2009 Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005)
- 10. What is Ratio-Dependence? (Dehaene, 2003; Gallistel & Gelman, 2000; Huntley-Fenner, 2001; Temple & Posner, 1998)
- 11. Approximate Number System Precision Representation of a number Low variability vs. High variability Target Number (Neider, 2004; Neider & Miller, 2003)
- 12. Approximate Number System Precision (Halberda, in press)
- 13. Weber’s Law Intensity: Maintaining a perception of brightness X 10 X 100
- 14. Weber’s Law and Number Discriminate 13 from 10 at 90% Discriminate 26 from 20 at 90%
- 15. Approximate Number System Precision Low Weber fraction= low variability and more precision High Weber fraction = high variability and less precision SD = n(w) Target Number (Halberda, in press; Neider, 2004; Neider & Miller, 2003)
- 16. The Linear Number Properties The log-to-linear shift is hard
- 17. Why Is Number Mapping Difficult? • Cardinality Principle • Put two fish in the pond (Butterworth, 2005; Carey & Sarnecka, 2006, 2008; Lipton & Spelke, 2006; Wynn, 1990)
- 18. Why Is Number Mapping Difficult? • Order Irrelevance Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
- 19. Why Is Number Mapping Difficult? • Abstraction Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
- 20. Why Is Number Mapping Difficult? • Stable Order Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
- 21. Why Is Number Mapping Difficult? • Successor Function Principle (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990) N + 1 1 2 3 4 5 6 7 8 N + 1
- 22. What is the Significance? • Developmental trajectory • Predictive validity • Natural ability • Dyscalculia • Early intervention
- 23. What Do I Want to Know? Hypothesis • Impact of visual modeling on number sense • Types of visual model(s) that are effective • Pre-symbolic intervention • Generalization Implications
- 24. The Study
- 26. Randomized Control Group Pretest/Post-Test Participants Preschool children: Mapping numbers No formal math instruction
- 27. How do We Measure Number Sense? Quantity Comparison Number-to-Position (Halberda et al., 2008; Siegler & Booth, 2004)
- 28. How Do We Measure Number Sense? Quantity Comparison Number-to-Position
- 29. Quantity Comparison: Sample Output
- 30. Quantity Comparison: Sample Output
- 31. Number-to-Position: Sample Output (Siegler & Booth, 2004)
- 32. Group 1 Group 1: Cardinality Counting tasks
- 33. Group 2 Group 2: Cardinality Relative Magnitude Number placement game
- 34. Group 3 Group 3: Cardinality Relative Magnitude Successor Function (N+ 1) Number placement and Space game +
- 36. Post-Testing • Where is 7?
- 37. Questions… “Children have their own preschool arithmetic, which only myopic psychologists could ignore.” ~Vygotski Thought and Language (1935, p. 84)
Editor's Notes
- This morning I will discuss the background and design for my proposed study: "Effect of Visual Modeling on Preschoolers' Number Mapping". The study is based on our natural number sense and its impact on the development of mathematical competence.
- This morning I will discuss the background and design for my proposed study: "Effect of Visual Modeling on Preschoolers' Number Mapping". The study is based on our natural number sense and its impact on the development of mathematical competence.
- Our natural number system was originally adapted for estimation, thus it is called the Approximate Number System. Evidence from infant studies, animal studies, brain studies, and studies of indigenous people who do not have number words tells us the ANS is common to all animals and people, it is functioning in early infancy, and it operates independently of language. In other words, the ANS has deep phylogenetic roots.
- The ANS is an ancient perceptual system. It adapted as an early warning system in fight or flight situations, foraging situations, and for survival in groups. In the wild, animals need to be able to instantly compare an opposing group’s size to the size of their own group. Animals must also be able to quickly find the most plentiful food sources. This is the difference between “having lunch and being lunch”. This is the central function of our ANS: To roughly compare groups.
- The approximate number system is more precise in some individuals than others. The precision of your Approximate Number System is the same as your “number sense”.
- Tokyo subway passengers using their ANS
- The ANS has several shared characteristics: 1) The system is cross-modal, meaning we can roughly gauge and compare quantities of sounds, objects, and actions
- The ANS has several shared characteristics: 2) The ANS does not track the exact number of objects in a set. It is used for gross estimation and comparison, approximate addition and approximate subtraction.
- The ANS has several shared characteristics: 3) We spontaneously map quantity to space For example, DeHevia and Spelke evaluated whether infants could map quantity to space. They presented infants with dot arrays of varying sizes and lines of varying lengths. After infants had been habituated to the stimuli (it was no longer novel to them--they lost interest), combinations of lines and dot arrays were presented. Infants preferred arrays and line lengths that were congruent. Preference was measured in look-time. In other words, we have a natural propensity to represent quantity on a linear number line. In the proposed study, I will be using quantity-to-space models in one of the treatment groups.
- The ANS has several shared characteristics: 4) The ANS operates on a logarithmic scale-- it is ratio-dependent The ANS is used to make rough quantity comparisons. We can quickly tell the larger of two sets as long as the ratio of one to the other is sufficient. Our precision with quantity discrimination improves over time. For example, 6 month olds can discriminate at a ratio of 1:2, and 9 month olds can discriminate at a ratio of 2:3. Adults have more precision, but there is a limit as the ratio of one set to the other approaches one. There is also individual variability in our ability to compare sets. In this study, we will be focusing on whether it is possible to improve an individual's quantity discrimination skill. Dot comparison 6 months:1:2 (8 vs. 16, 16 vs. 32) (Lipton & Spelke, 2003; vanMarle & Wynn, 2006) 9 months:2:3 (8 vs. 12, 16 vs. 24) (Lipton & Spelke, 2003; McCrink & Wynn, 2005; Strauss & Curtis, 1981; Wood & Spelke, 2005; Xu & Arriaga, 2007)) Adults: 9:10 (Halberda, Mazzocco, & Feigenson, 2008) This study will focus on the characteristics of quantity-space mapping and logarithmic scaling
- The ANS is logarithmic. This means that pre-numerate children think about quantity in ratios. If we were to represent a logarithmic scale on a number line, equal magnitude differences would be the same distance apart. For example, the linear distance between 4 and 8 is twice the distance between 2 and 4, but their logarithmic distances, or ratios, are equal (1:2). Because young children think about quantity in ratios, they mentally exaggerate the distances between numbers on the lower end of their mental number line and compress the distances between numbers on the higher end. Compressed scaling makes discrimination difficult when numbers are close together in quantity or large; that is, farther up the number line (Dehaene, 2003). At some point in development, children have to take the logarithmic scale and "map" it onto an equal-interval, or linear scale (see figure).
- Numbers derive meaning from their positions relative to one another. It is for this reason that number sense is best measured by indexing a person's accuracy in comparing sets. The ANS evolved to do just that, but in an approximate fashion. There is error in our ability to represent quantity. This is not a theoretical model. Neider and Miller (2003) isolated numerosity-selective neurons in primates and measured neural activity as the primates were presented with dot arrays. Primates were able to match dot arrays to key presses, but with approximate or "noisy" brain activations (Neider, 2004). They found that the neural activation associated with a given number clusters around that number as it's mean, with decreasing activation on either side of the target number. For example, if a target array has five items, brain activation is highest at a numerosity of five, less intense at four and six, and even less intense at three and seven (Halberda, in press; Neider & Miller, 2003). The firing activity of neurons has an inherent variability, called "noise" (Gallistel & Gelman, 2000).
- If we placed these distributions along a mental number line, there would be overlap between adjacent distributions. Moving up the scale distributions become wider and the resulting overlap between numerosities is greater (Halberda, in press; Neider, 2004). Numbers that are far apart are easier to discriminate because there is less overlap between the distributions of possible guesses. Smaller numbers are easier to discriminate because of the limited range of possible activations. However, it is the ratio between sets, rather than number position, that drives discriminability. For example, the numbers 4 and 5 have the same discriminability as the numbers 16 and 20, because they differ by equal ratios. The distributions for 4 and 5 have less spread, but 16 and 20 are far enough apart that their larger standard deviations result in the same amount of overlap (Halberda, in press).
- Ratio-dependent performance operates in several areas of perception, including brightness, weight, and temperature. We call this phenomenon Weber's Law. A person requires a certain magnitude of difference between two numbers to discriminate them, and this difference is a constant. The amount of variability in a person's number representations equals each target number multiplied by the Weber fraction for that person. I will be taking a Weber fraction for each participant’s number sense, and then again at post-testing to determine if there has been a change in their ANS precision.
- Mapping number is hard. Moving from a ratio scale to a linear scale (exact quantity) takes several years. For example, children begin to use counting words at about 2 years old, but they do not understand cardinality, or the idea that the last number in a count sequence equals the exact number in a set, until about age 5. Most teachers model cardinality through counting sets of objects. Our control group will be counting objects. 2.5 Recite count words: “one” means 1, “two” means “a bunch” 3 “Two” means 2, “three” means something more than 2 3.5 “Three” means 3 4 Each count word is an exact amount: Cardinality 5 Number is infinite: N + 1 or Successor Function
- Other principles of number children need to know: Order Irrelevance Principle: We can start counting with any object, and as long as we count each object only once, we will get the correct amount
- Other principles of number children need to know: Abstraction Principle: We can have three baseballs or three wishes, three idiosyncrasies, or three dreams
- Other principles of number children need to know: Stable Order Principle: Sometimes order does matter. Seven is a big or small number depending on the scale. In other words, numbers derive their meaning from their relative position
- Other principles of number children need to know: Successor function: The idea that number is infinite and generative. N + 1. Each number includes itself and everything that came before it. In this study, we will be modeling Cardinality, Relative Magnitude, and Successor function to determine what, if any impact these models may have on ANS precision (number sense).
- Why should we care about the mapping problem? The relationship between ANS precision and math achievement persists throughout development (Halberda, Mazzocco, & Feigenson, 2008; Mazzocco, Feigenson, & Halberda, 2011) ANS precision is a better predictor of math success than IQ, reading, rapid naming, and spatial skills (Halberda Mazzocco, & Feigenson, 2008; Mazzocco, Feigenson, & Halberda, 2011) Children who have learned to count can harness their ANS skills to solve symbolic arithmetic problems, even without formal instruction (Barth, LaMont, Lipton, & Spelke, 2005 ; Gilmore, McCarthy & Spelke, 2007) ANS precision can differentiate between children with dyscalculia and “poor achievers” (Halberda, Mazzocco, & Feigenson, 2008; Mazzocco, 2010) Dyscalculia occurs with the same frequency as dyslexia, but there is no known intervention (Butterworth, 2002, 2010; Geary, 2004, 2010) The ANS is operating at birth, so it may be possible to intervene early, prior to formal instruction in numerals
- Can ANS Precision (number sense) be improved with visual modeling? If so, what kind(s) of visual models are effective? If children improve the linearity of their number line estimation, will this result in improved ANS precision? Does visual modeling impact generalization of number representation from scale to scale?
- The study I have designed is a randomized control group pretest-posttest design. . In this design, the conditions for experimental and control groups are identical, except that experimental groups participate in a treatment. There are three groups, one control and two treatment, in this experiment. The treatments for each group are categorical independent variables. Each participant will take four brief assessments: (1) two covariate or pre-test measures and (2) two dependent or post-test measures.
- As we have discussed, the accuracy of our number comparison judgments is the key component in ANS precision (number sense). It is for that reason that the best measure of ANS precision is through quantity comparison tasks. Let's try a quick example of one of the tasks I will be using [go to one-trial simulation] Another way to measure ANS precision is through numberline estimation. The participant is given a blank, unmarked number line and they are asked to locate target numbers on the line.
- Because the ANS is operating before children have exposure to numerals and because it is so central to future math performance, I want to know if it is possible to accelerate and improve ANS precision early, before formal instruction in math. So far, there are no studies that use non-symbolic stimuli. For the proposed study, I am using a preschool version of the task you just tried, and I have also adapted the Number-to-Position task for non-symbolic stimuli (see figure).
- Here is an output chart from an 11 year old on Panamath.
- Here is the same 11 year old. Remember that the precision of a person's estimates is described by the Weber fraction, which indicates the amount of "noisiness" in your underlying number sense representations. A larger Weber fraction means more variability in the estimates (a larger SD) and a small Weber fraction means less variability (a smaller SD). In this example, the weber ratio of an 11 year old has been plotted against performance at the 10th and 90th percentiles.
- On the number-to-position or numberline estimation task, students' estimates can be compared to a best-fit trend line. Here is an example from Siegler & Booth 2004. When children were asked to plot numbers on a blank numberline, the Kindergarten group produced logarithmic representations, the first graders produced mixed representations, and the second graders produced mostly linear results.
- 1) The control group will be exposed to typical preschool activities such as using counters. The focus here is on exact cardinality, or the total number of objects in a set.
- 2) Group 2 will play a linear number game using manipulatives that demonstrate placement of quantities on a number line as well as the relative size of those numbers, or their relative magnitude.
- 3) Group 3 will play a similar game, but they will also be exposed to a model of the continuous (successor function) nature of numbers and the mapping of quantity to space. To so this, I will use an adapted manipulative that unfolds to show the length of the number along the number line.
- Figure 1: The preschool version of Panamath quantity comparison task Figure 2: My nonsymbolic version of the Number-to-position task by Siegler & Booth (2004)
- In previous numberline estimation studies by Siegler, Booth, Opfer, Ramani, and others, children were able to improve the linearity of their numberline estimations when given some anchor numbers. For example, on a 1-100 scale children could produce linear representations after being shown the locations for 25, 50, and 75. The researchers claimed that this operation improved number sense. However, I have two questions about this: 1) The children were unable to generalize by producing linear representations on a larger scale (eg. 1-1000) and 2) The children were never tested on an independent criterion measure or "pure" measure of number sense such as Panamath. If anchor cues improve number sense, then those cues should improve performance on a quantity comparison task such as Panamath. I want to find out if visual modeling of number properties produces this type of generalizability and if so, what type.